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The Classical Hall effect

The Classical Hall effect. Reminder: The Lorentz Force. F = q [ E + ( v  B )]. Lorentz Force: Review. Velocity Filter Undeflected trajectories in crossed E & B fields: v = (E/B) Cyclotron Motion F B = ma r  qvB = (mv 2 /r) Orbit Radius

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The Classical Hall effect

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  1. The Classical Hall effect

  2. Reminder: The Lorentz Force F = q[E + (v B)]

  3. Lorentz Force: Review • Velocity Filter • Undeflected trajectories in crossed E&Bfields: v = (E/B) • Cyclotron Motion • FB = mar  qvB = (mv2/r) • Orbit Radius • r = [(mv)/(|q|B)] = [p/(q|B|]) • Orbit Frequency • ω = 2πf = (|q|B)/m • Orbit Energy • K = (½)mv2 = (q2B2R2)/2m A Momentum Filter!! A Mass Measurement Method!

  4. Standard Hall Effect Experiment Setup Current from the applied E-field e+ v e- v Lorentz force from the magnetic field on a moving electron or hole E field Top view: Electronsdrift from back to front e- leaves + & – charge on the back & front surfaces Hall Voltage The sign is reversed for holes

  5. t d I I Electrons Flowing With No Magnetic Field Semiconductor Slice _ +

  6. When the Magnetic Fieldis Turned On .. I B-Field qBv

  7. As time goes by... High Potential I Low Potential qE qBv = qE

  8. Finally... V H I B-Field

  9. The Classical Hall Effect 1400 Hall Resistance Rxy(Ohms) Slope Related to RH & Sample Dimensions 1200 1000 800 600 400 ly 200 0 Ax 4 0 2 6 8 10 Magnetic Field (Tesla) lyis the transverse width of the sample Axis the transverse cross sectional area The Lorentz Force tends to deflect jx. However, this sets up an E-field which balances that Lorentz Force. Balance occurs when Ey = vxBz = Vy/ly. But jx = nevx(or ix = nevxAx). So Rxy = Vy / ix = RH Bz × (ly/Ax) whereRH = 1/ne

  10. Semiconductors:Charge Carrier Density via Hall Effect • Why is the Hall Effect useful? It can determine the carrier type (electron vs. hole) & the carrier density nfor a semiconductor. • How? Place the semiconductor into external B field, push current along one axis, & measure the induced Hall voltageVH along the perpendicular axis. The following can be derived: • Derived from the Lorentz force FE = qE = FB = (qvB). n = [(IB)/(qwVH)] Hole Electron + charge – charge

  11. The 2D Hall Effect The surface current density is sx = vxσ q, where σ is the surface charge density. Again, RH = 1/σe. However, now: Rxy = Vy / ix = RH Bz. since sx = ix /ly& Ey = Vy /ly. So, Rxydoes NOT depend on the shape of the sample. This is a very important aspect of the Quantum Hall Effect

  12. The Integer Quantum Hall Effect Very important: For a 2D electron system only First observed in 1980 by Klaus von Klitzing Awarded 1985 Nobel Prize. Hall Conductance is quantizedin units ofe2/h,or Hall ResistanceRxy = h/ie2, where iis an integer. The quantum of conductanceh/e2 is now known as the "Klitzing“!! Has been measured to 1 part in 108

  13. The Fractional Quantum Hall Effect The Royal Swedish Academy of SciencesawardedThe 1998 Nobel Prize in Physicsjointly to Robert B. Laughlin (Stanford), Horst L. Störmer (Columbia & Bell Labs) & Daniel C. Tsui, (Princeton) • The researchers were awarded the Nobel Prize for discovering that electrons acting together in strong magnetic fields can form new types of "particles", with charges that are fractions of electron charges. • Citation: “For their discovery of a new form of quantum fluid • with fractionally charged excitations” • Störmer & Tsui made the discovery in 1982 in an experiment using extremely powerful magnetic fields & low temperatures. Within a year of the discovery Laughlin had succeeded in explaining their result. His theory showed that electrons in a powerful magnetic field can condense to form a kind of quantum fluid related to the quantum fluids that occur in superconductivity & liquid helium. What makes these fluids particularly important is that events in a drop of quantum fluid can afford more profound insights into the general inner structure dynamics of matter. The contributions of the three laureates have thus led to yet another breakthrough in our understanding of quantum physics & to the development of new theoretical concepts of significance in many branches of modern physics.

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